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# problems

			       Problems

From "How to Select and Use a Calculator" by P.R. Jamale

0.000123 / 456 * 7890000 = 2.1282236

see if
\pi ~= 355 / 113
e ~= 2721 / 1001

see what a calculator does with:

(\v/123456789)^2
80 / 81			(less interesting these days)

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From "HP-45 Applications Book" by Hewlett-Packard (HP 00045-90320 Rev
B Reorder 00045-66001, Dec 1974)

find x such that x = e^(-x)				x ~= 0.567143290
find x such that 4 = x - 1/x				x ~= 4.236067978
find x such that x^x = 1000				x ~= 4.555535705
find x such that (x^x)^x = 1E100			x = 10
find x such that x^(x^x) = 1E100			x ~= 3.830482864
find x such that x^2+4*sin(x)=0, x in radians,		x ~= -1.933753764
find x such that x^3 = 3^x, 1 < x < e			x ~= 2.478052679

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Nifty formula (Mach formula) in the HP-91A Owner's Manual, page 73.

------------------------------------------------------------------------
/[({[(       [ 350 ]2)3.5   ][    (        )     ]-5.2656}    )0.286   ]
/5[({[(1 + 0.2[-----] )   - 1][1 - (6.875e-6)25500]       } + 1)     - 1]
\/  [({[(       [661.5] )      ][    (        )     ]       }    )        ]

in solver form:

SQRT(5 * ((((1 + 0.2 *(350 / 661.5)^2)^3.5 - 1) *
(1 - 6.875e-6 * 25500)^-5.2656 + 1)^0.286 - 1))

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Miscellaneous formulae in HP95LX "HPCALC" format, also suitable for most
of the calculators with algebraic solvers.  By Craig Finseth

19B-style date arithmetic:

{19B Date|0=DDAYS(DATE1,DATE2,if(s(DAYS),1,if(s(x360D),3,2)))-
if(s(DAYS),DAYS,if(s(x360D),x360D,x365D))+0*TODAY*l(TODAY,CDATE)}

Some of the above problems:

{x^3=3^x}
{4=x-INV(x)}
{x^x=1000}
{x=EXP(-x)}
{x^(x^x)=big}
{(x^x)^x=big}

Fun things to graph:

{x^3=3^x}
{sin(x)/x}
{sin(x)}
{SQ(x)+4*SIN(x)=0}

For testing solvers:

{TVM|(1-EXP(-n*LNP1(i/100)))*PMT*100/i+PV=-FV*EXP(-n*LNP1(i/100))}
{sinx=SIGMA(i,0,last,1,(-1)^i*x^(2*i+1)/FACT(2*i+1))}
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Comp.sys.handhelds:;

... It's certainly accurate enough that if you do my favourite
calculator test, it gives a reasonable answer.  The test is:

sqr(asin(acos(atan(tan(cos(sin(0.01**2)))))))

And the result was (should be) 0.0104586394192 (on a Psion-3).